We know that the definition of integration measure states that 1) $\tilde{\mu}(I_A) = \mu(A)$ 2) $\tilde{\mu}(aA+bB) = a\mu(A) + b\mu(B)$
Is it true that $\tilde{\mu}(I_A - I_B) = \tilde{\mu}(I_A) - \tilde{\mu}(I_B) = \mu(A) - \mu(B)$? Would this still hold if it was absolute $|I_A - I_B|$ ?
$\int (I_A-I_B)=\int I_A -\int I_B$ holds if either $\mu (A) <\infty$ or $\mu (B) \infty$. If both measures are infinity then $\mu (A)-\mu(B)$ is not even defined. I is not true that $\int |I_A-I_B|=|\int I_A -\int I_B|$. In fact $\int |I_A-I_B|=\int I_{A\setminus B} +\int I_{B\setminus A}$.